Methods and apparatus for predicting convergence of a transversal waveform relaxation algorithm

ABSTRACT

Methods and apparatus are provided for predicting the number of iterations needed for a computed Transversal Waveform Relaxation solution to achieve a given level of accuracy. In this manner, a Transversal Waveform Relaxation algorithm is disclosed that provides full automation. According to one aspect of the invention, a circuit is analyzed having transmission lines. One or more transmission line parameters of the circuit are obtained, as well as the intrinsic behavior, E(ω), and strength of coupling, N(ω), of each of the transmission lines. In addition, a relative error bound is obtained for the circuit based on the intrinsic behavior, E(ω), and strength of coupling, N(ω), of the transmission lines and a predefined error threshold. The process then iterates until the relative error bound satisfies the error threshold.

FIELD OF THE INVENTION

The present invention relates to the analysis of electrical circuits,and more particularly, to the analysis of transmission line circuitsusing Transversal Waveform Relaxation techniques.

BACKGROUND OF THE INVENTION

Electrical circuits with transmission lines are typically analyzed toensure proper functioning of the circuits. The coupling between multiplelines and the resultant coupled signals are an important aspect of thesetransmission line circuits. Power distribution systems, for example,often must be analyzed for stability and other properties. Similarly, ininstrumentation circuits and computer circuitry in racks or cabinets thenoise coupled between transmission lines needs to be understood andminimized.

A number of techniques have been proposed or suggested for analyzingmultiple wire transmission lines. Such techniques are described, forexample, in Clayton Paul, Analysis of Multiconductor Transmission Lines,Ch. 5 (Wiley, 1994). While these techniques are suitable for theanalysis of models with a few lines, the complexity increases rapidly asthe number of lines increases. Some simplified techniques have beenproposed to approximate the solution for many transmission lines withonly neighbor-to-neighbor wire coupling. These approaches are suitablewhere reduced accuracy is acceptable to gain speed.

Existing techniques for analyzing multiple wire transmission lines arelimited in the number of coupled lines or wires that can be analyzedsimultaneously. The complexity of the coupling calculation increasesrapidly as the number of lines increases, and the accuracy of theresults decreases with the increasing number of lines. Hence, theexisting techniques are unable to handle a large number of lines due toexcessive computation time and the results become questionable. Someprior art techniques ignore the couplings for more than two lines tospeed up the process. Other techniques are based on having only linearcircuits to speed up the calculation process and are thereforeunsuitable for handling even typical transmission line circuits, whichinclude surrounding nonlinear drivers and receivers.

U.S. patent application Ser. No. 10/776,716, entitled “System and MethodFor Efficient Analysis of Transmission Lines,” incorporated by referenceherein, discloses “Transversal Waveform Relaxation” techniques foranalyzing multiple wire transmission lines by determining which sourcesinfluence each of a plurality of transmission lines, based on couplingfactors. Transmission line parameters are computed based on the sources,which influence each transmission line. A transient or frequencyresponse is analyzed for each transmission line by segmenting each lineto perform an analysis on that line. The step of analyzing is repeatedusing waveforms determined in a previous iteration until convergence toa resultant waveform has occurred. For a more detailed discussion ofsuch Transversal Waveform Relaxation techniques, see, for example,Nakhla et al., “Simulation of Coupled Interconnects Using WaveformRelaxation and Transverse Partitioning,”. EPEP'04, Vol 13, pp 25-28,Portland, Oreg., October 2004, incorporated by reference herein.

While such Transversal Waveform Relaxation techniques have greatlyimproved the analysis of multiple wire transmission lines, they sufferfrom a number of limitations, which if overcome, could provide furtherimprovements. For example, the Transversal Waveform Relaxation is notfully automated and requires some manual input to determine when thealgorithm has achieved a given level of accuracy. The full automationhas been hampered by the absence of a quantitative link between thephysical characteristics of a multiconductor transmission line system,as expressed by the per-unit-length capacitance, resistance, inductanceand conductance matrices, and the convergence behavior of a TransversalWaveform Relaxation algorithm when used for the electrical analysis oflarge scale multiconductor transmission line systems. It has beenqualitatively observed that when the electromagnetic coupling betweenthe lines in the system is weak, the Transversal Waveform Relaxationalgorithm needs only a few iterations to converge to an accuratesolution.

A need therefore exists for methods and apparatus for predicting thenumber of iterations needed for the computed solution to achieve a givenlevel of accuracy. A further need exists for a Transversal WaveformRelaxation algorithm that provides full automation.

SUMMARY OF THE INVENTION

Generally, methods and apparatus are provided for predicting the numberof iterations needed for a computed Transversal Waveform Relaxationsolution to achieve a given level of accuracy. In this manner, aTransversal Waveform Relaxation algorithm is disclosed that providesfull automation. According to one aspect of the invention, a circuit isanalyzed having transmission lines. One or more transmission lineparameters of the circuit are obtained, as well as the intrinsicbehavior, E(ω), and strength of coupling, N(ω), of each of thetransmission lines. In addition, a relative error bound is obtained forthe circuit based on the intrinsic behavior, E(ω), and strength ofcoupling, N(ω), of the transmission lines and a predefined errorthreshold. Thereafter, the process iterates until the relative errorbound satisfies the error threshold.

The transmission line parameters of the circuit may include one or moreof capacitance, resistance, inductance and conductance. The intrinsicbehavior, E(ω), indicates the single line bound of each transmissionline and the strength of coupling, N(ω), measures the size of thecoupling terms between two or more transmission lines.

A more complete understanding of the present invention, as well asfurther features and advantages of the present invention, will beobtained by reference to the following detailed description anddrawings.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a block diagram illustrating a system in accordance with anembodiment of the present invention;

FIG. 2 is a flow diagram illustrating a method for transmission lineanalysis using a transverse waveform relaxation process;

FIG. 3 depicts an illustrative geometry for a multiple transmissionlines to be analyzed in accordance with the present invention; and

FIG. 4 is a flow chart describing an exemplary Transversal WaveformRelaxation process that incorporates a stopping criterion in accordancewith the present invention.

DETAILED DESCRIPTION

The present invention provides methods and apparatus for determining anumber of iterations needed for a solution computed by a TransversalWaveform Relaxation algorithm to achieve a given level of accuracy.

FIG. 1 is a block diagram illustrating a system 100 in which the presentinvention can operate. As shown in FIG. 1, an exemplary system 10includes a computer 12, such as a personal computer or a mainframe.Computer 12 includes any interface devices known in the art. Computer 12may include a plurality of modules or software packages that may beresident in the system or coupled thereto via a network or the like. Forexample, computer 12 may be provided access to electronic designautomation (EDA) libraries or other circuit databases 14, which includeelectrical circuits or integrated circuit chip designs.

A module 16 may include one more programs or subroutines for carryingout methods in accordance with the present invention. Module 16 mayinclude transmission line analysis programs, including a solver 17 orcode to determine coupling factors, perform sliding calculations, updatecoupling models and perform transient analysis, among other things aswill be described in greater detail herein below. Module 16 may beincorporated into other programming packages, such as full-blown circuitanalysis systems or programs. In addition, as discussed further below inconjunction with FIG. 4, the module 16 includes a process 400 fordetermining a number of iterations needed for a solution computed by thetransmission line analysis programs 200 (FIG. 2) to achieve a givenlevel of accuracy.

A computer aided design (CAD) module or program 18 may be included toimport designs or design information to the system 10 to provide theappropriate circuit analysis. CAD schematics and or EDA data fromdatabase 14 may be employed as inputs to module 16 to analyze componentsof a design, and preferably transmission lines in the design.

FIG. 2 is a flow diagram illustrating a method 200 for transmission lineanalysis using a transverse waveform relaxation process. Thetransmission line analysis process 200 may be referred to as atransverse waveform relaxation process. In circuit designs, one or moretransmission lines may be present. To handle a plurality of coupledlines, the impact of each neighboring transmission line needs to beconsidered. For a more detailed discussion of suitable transversewaveform relaxation processes, see, for example, U.S. patent applicationSer. No. 10/776,716, entitled “System and Method For Efficient Analysisof Transmission Lines,” or Nakhla et al., “Simulation of CoupledInterconnects Using Waveform Relaxation and Transverse Partitioning,”EPEP'04, Vol 13, pp 25-28, Portland, Oreg., October 2004, eachincorporated by reference herein.

Generally, the transversal waveform relaxation algorithm 200 initializesthe electrical states of all wires; analyzes every wire at a time withcouplings from all other wires implemented with known voltage andcurrent sources; and iterates until convergence. The present inventionprovides techniques for determining when such convergence has occurred.

The exemplary transverse waveform relaxation process 200 shown in FIG. 2addresses neighbor-to-neighbor coupling early on in the process. Thiskeeps the problem sparse if the model includes a large number of coupledtransmission lines. Also, a coupling model of parameters, which have tobe computed at the same time, is limited to a relatively small number.The model limits the number of coupled sources needed to represent thecouplings. Then, each wire is individually analyzed as a subcircuitwhile the coupled sources are taken into account, and the waveformsbetween the last two iterations are compared to check for convergence.

In block 202, coupling factors are determined for transmission lines ina given design. Coupling factors are determined by calculating theinfluence of neighboring lines on a given line. In one embodiment, aninductance coupling factor (cf) may be calculated as:${cf} = \frac{L_{12}}{\sqrt{L_{11}L_{22}}}$where L₁₂ is the inductive influence of L₂ on L₁ (coupling term) and L₁₁is the inductance of line 1 and L₂₂ is the inductance of line 2 (thesemay be referred to as self terms). A similar coupling factor can becalculated for resistance$\left( {{cf} = \frac{R_{12}}{\sqrt{R_{11}R_{22}}}} \right)$and capacitance$\left( {{cf} = \frac{C_{12}}{\sqrt{C_{11}C_{22}}}} \right).$Other coupling factors may be used and preferably dimensionlessvariables. Also, default values, e.g., 0.001, may be inserted into theprogram for coupling factors to ensure a nonzero number exists in thecase where the coupling factor is not available or for other reasons.

The coupling factors are estimated first with approximate computations.This leads to the knowledge of how many transmission lines are to beincluded in each segment of the calculation. The segments are computedin an overlapping way such that all the interactions are taken intoaccount so that each line can be analyzed individually taking intoaccount the pertinent couplings for each line.

After calculating the coupling factors for transmission lines in adesign, the coupling factors are compared to a threshold value(s), inblock 204, to determine if they will have an influence on neighboringlines. For example, L_(coupling)<L_(tolerance);C_(coupling)<C_(tolerance); and R_(coupling)<R_(tolerance).

The tolerance or threshold is preferably set by a designer but can alsobe calculated based on parameters or criteria for a given design. Forexample, in sensitive equipment, a smaller tolerance may be neededmeaning smaller influences should be considered in analyzingtransmission line parameters. Coupling factors that are determined to betoo small may be disregarded in future calculations for a given segment.However, since many circuits are dynamic and different portions of acircuit may be operational at a different time, different time framesmay be investigated to ensure a complete solution.

For example, for a given line, coupling factors are employed todetermine the influence of other lines on the line in question. Based onthese estimates, the calculations are segmented for each line to includethe most influential coupling effects.

In block 206, a sliding computation is optionally performed to calculatetransmission line parameters (L, C and R), for example, if all of thetransmission line parameters cannot be obtained in accordance with step202. These calculations are preferably based on geometric features. Thecalculation of the per-unit line parameters L, R, and C is preferablyperformed in a segmented way, since the simultaneous calculation ofthese matrix quantities is very expensive for more than a few lines.Hence, each segmented calculation will include the computation of the L,R, C parameters for a number of lines. The number of steps needed todetermine these parameters is much smaller than the total number oflines since the most influential lines are considered. This simplifiesthe evaluation of the L, R, C parameters since each sub-problem is muchsmaller than the large single evaluation of each parameter.

For example, assuming 100 transmission lines, the coupling evaluationdetermines that 5 lines should be included near each line to accuratelytake the coupling into account. So for the sliding calculation, thefirst 15 lines are evaluated simultaneously using, for example, astandard field solver for L, R and C. This result can be used for thefirst 10 lines in the transverse waveform relaxation (WR) approach givenherein. Then, in a next sliding field calculation, the next L, R, C, areevaluated for 15 lines from line 10 to line 25. Then, this calculationcan be used to evaluate the transverse WR for lines 11 to 20, and so on.Hence, the field calculation is completed only on a subset of the lineswhich is much faster since the compute time of the field solverincreases enormously with the number of lines considered.

In one embodiment, the characteristics of a circuit as defined in a CADschematic are employed to make these sliding calculations. The slidingcalculations provide a baseline for the transient analysis as will bedescribed hereinafter.

Based on the sliding computation, in block 208, a coupling model ormodels are employed to reduce the circuit characteristics into terms ofvoltage and/or current sources with lumped elements (L, R, C) oralternately uses the method of characteristic models to model thecircuit.

In block 210, a transient analysis/frequency domain analysis of thetransmission lines is performed preferably one wire at a time. Thetransient analysis/frequency domain analysis is based on a transmissionline response to surrounding circuits using coupled sources to othercoupled lines as provided by the models set forth in block 208.

In one embodiment, partitioning along the coupling of the lines isperformed. In other words, each line is taken one at a time consideringthe most pertinent coupling influences on that line. Alternately,partitioning over the length of the line may be performed as well or inaddition to a calculation for the partitioning of the coupling of theline.

In block 212, the transient analysis of block 210 is repeated untilconvergence is achieved by comparing a previous value of the waveformsdetermined by the transient analysis from a previous iteration to thewaveforms determined in the present iteration. If convergence isachieved the resultant waveforms have been determined and are availablein block 214. If convergence has not yet been achieved, then the programreturns to block 210 to recalculate the waveforms.

As previously indicated, the present invention provides methodsdiscussed further below in conjunction with FIG. 4 to determine thenumber of iterations needed for a computed solution to achieve a givenlevel of accuracy.

FIG. 3 depicts an illustrative geometry 300 for a multiple transmissionlines to be analyzed in accordance with the present invention.Transmission lines 302 are numbered 1 to N in the depicted section of acircuit 300. Lines 302 are marked with an A to indicate that they areaggressor lines. These lines are exited with some external circuitry. Incontrast, the lines which are marked with a V are victim lines which arenot excited with external sources.

Using one method, e.g., set forth with reference to FIG. 2, thesubcircuits/lines of FIG. 3 are analyzed starting at line 1, insequence, until line N is reached. This sequence is followed for eachcalculation in FIG. 2. Then, the sequence is repeatedly followed gountil convergence in the transient/frequency analysis (e.g., blocks 210and 212).

A more efficient method is based on signal flow. For example, initially,all coupled waveform sources are set to zero. Then, starting with theanalysis of the circuits, which include the aggressors (A) first, newcoupled-source quantities are available from the coupling model (block206). Then, the nearest neighbors are analyzed since they will includethe largest signals next to the aggressors. The process progressesthrough all the wires until all of the wires have been visited. In eachstep/iteration, the latest, updated waveforms are employed.

These methods are directly applicable to parallel processing for circuitproblems, which include transmission lines. Each of the N transmissionlines forms a separate subsystem with a transverse decoupling scheme(e.g., portioning along coupling lines or effects). Further partitioningis possible along the length of the line using conventional techniques.

Partitioning or segmenting line by line (coupling) leads to 2Nsubsystems which can be analyzed on separate processors where the onlyinformation that needs to be exchanged between processors is waveforms.Hence, an enormous gain in speeding up the process by parallelprocessing is achieved.

Convergence Criterion

As previously indicated, the present invention provides methods andapparatus for determining a number of iterations needed for a solutioncomputed by a Transversal Waveform Relaxation algorithm to achieve agiven level of accuracy. As discussed hereinafter, the present inventionrelates the number of iterations needed to achieve a certain level ofaccuracy to an error threshold. In addition, the present inventionquantifies the physical intuition that the weaker the couplings betweenthe lines, the faster the convergence.

The disclosed techniques relate the convergence properties to input dataof the transmission line system, such as per-unit-length R, L, G, Cmatrices. An automatic stopping criterion is provided for the iterativeprocess that allows the Transversal Waveform Relaxation software to berun by users who need not be skilled in the art.

As discussed in the following sections, the present invention is basedon a transformation of the Transversal Waveform Relaxation algorithm asapplied to the well-known telegrapher's equations of multiconductortransmission lines into an iterative process applied to an integralequation, referred to as a Volterra integral equation. Thereafter,formulas for the convergence rate of Transversal Waveform Relaxation andthe approximation error incurred, if the iterative process is stopped,are derived. The Transversal Waveform Relaxation process 200 of FIG. 2can then be augmented with an error threshold and the ability to computethe error as a function of the iteration number. Automatic terminationis achieved when the error threshold is reached.

The Telegrapher's Equation

The set of first-order partial differential equations describing thebehavior of the multiconductor transmission lines are: $\begin{matrix}{{{\frac{\partial}{\partial x}{v\left( {x,t} \right)}} = {{- {{Ri}\left( {x,t} \right)}} - {L\quad\frac{\partial}{\partial t}{i\left( {x,t} \right)}}}}{{\frac{\partial}{\partial x}{i\left( {x,t} \right)}} = {{- {{Gv}\left( {x,t} \right)}} - {C\quad\frac{\partial}{\partial t}{v\left( {x,t} \right)}}}}} & (1)\end{matrix}$where R, L, G, Cε

^(N×N) are the per-unit-length (PUL) matrices of the N lines. Thesematrices are symmetric positive semidefinite. Each of the matrices issplit between a diagonal part and non-diagonal part. The diagonal partwill be denoted with D_(X) and the nondiagonal part with N_(X) where Xrefers to the original matrix as follows:R=D _(R) +N _(R)  (2)L=D _(L) +N _(L)G=D _(G) +N _(G)C=D _(C) +N _(C.)The D matrices are the self parameters of the lines while the N matricesare the coupling parameters. The following notation is used:D _(Z)(ω)=D _(R) +jωD _(L)  (3)N _(Z)(ω)=N _(R) +jωN _(L)D _(Y)(ω)=D _(G) +jωD _(C)N _(Y)(ω)=N _(G) +jωN _(C)where jω is the Fourier variable. Using the above matrix splittings, thetelegrapher's equation can be written in the time domain as$\begin{matrix}{{{\frac{\partial}{\partial x}{v\left( {x,t} \right)}} = {{{- D_{R}}{i\left( {x,t} \right)}} - {D_{L}\quad\frac{\partial}{\partial t}{i\left( {x,t} \right)}} - {N_{R}{i\left( {x,t} \right)}} - {N_{L}\frac{\partial}{\partial t}{i\left( {x,t} \right)}}}}{{\frac{\partial}{\partial x}{i\left( {x,t} \right)}} = {{{- D_{G}}{v\left( {x,t} \right)}} - {D_{C}\quad\frac{\partial}{\partial t}{v\left( {x,t} \right)}} - {N_{G}{v\left( {x,t} \right)}} - {N_{C}\frac{\partial}{\partial t}{v\left( {x,t} \right)}}}}} & (4)\end{matrix}$and in the frequency domain as $\begin{matrix}{{{\frac{\partial}{\partial x}{V\left( {x,\omega} \right)}} = {{{- {D_{Z}(\omega)}}{I\left( {x,\omega} \right)}} - {{N_{Z}(\omega)}{I\left( {x,\omega} \right)}}}}{{\frac{\partial}{\partial x}{I\left( {x,\omega} \right)}} = {{{- {D_{Y}(\omega)}}{V\left( {x,\omega} \right)}} - {{N_{Y}(\omega)}{V\left( {x,\omega} \right)}}}}} & (5)\end{matrix}$

The above two first-order equations can be combined in a single firstorder equation of order 2N: $\begin{matrix}{{{\frac{\partial}{\partial x}{\Phi\left( {x,\omega} \right)}} = {{{- {D(\omega)}}{\Phi\left( {x,\omega} \right)}} - {{N(\omega)}{\Phi\left( {x,\omega} \right)}}}}{{where}\text{:}}} & (6) \\{{{\Phi\left( {x,\omega} \right)} = \begin{bmatrix}{V\left( {x,\omega} \right)} \\{I\left( {x,\omega} \right)}\end{bmatrix}},{{D(\omega)} = \begin{bmatrix}O & {D_{Z}(\omega)} \\{D_{Y}(\omega)} & O\end{bmatrix}},{{N(\omega)} = \begin{bmatrix}O & {N_{Z}(\omega)} \\{N_{Y}(\omega)} & O\end{bmatrix}}} & (7)\end{matrix}$

Equation 7 describes a linear system with a matrix D(ω) and an inputmatrix N(ω) through which the electromagnetic couplings impact theelectrical states of the wires. Note that if N(ω) is the zero matrix, Nindependent single-line telegrapher's equations are obtained. Thegeneral solution of Equation 7 is given byΦ(x,ω)=e ^(−D(ω)x)Φ(0,ω)−∫^(x) ₀ e ^(−D(ω)(x−ζ)) N(ω)Φ(ζ,ω)dζ  (8)

If the initial vector Φ(0,ω) (voltages and current at the near endports) were known, the above equation would have been, for the variablex and at each frequency ω, a Volterra integral equation of the secondkind. This type of integral equations can be solved according to thePicard iteration.Φ^((r+1))(x,w)=e ^(−D(ω)x)Φ(0,ω)−∫^(x) ₀ e ^(−D(x)(x−ζ))N(ω)Φ^((r))(ζ,ω)dζ.  (9)

Note that this equation is almost identical to Equation (6) in Nahkla etal., rewritten here asΦ^((r+1))(x, ω)=e ^(−D(ω)x)Φ^(r+1))(0,ω)−∫^(x) ₀ e ^(−D(ω)(x−ζ))N(ω)Φ^((r))(ζ,ω)dζ.  (10)

The main difference between Equation 9 and Equation 10 is in theinitial-condition term where it is kept at its original value in thePicard iteration (it is a real initial condition) but is updated in thewaveform relaxation, using the boundary conditions of each transmissionline. The value of Φ(0,ω) is not known a priori but depends on thenear-end and far end-end terminations of the transmission line. It isnoted that the most natural initial guess for starting the TWR iterationin the transmission line case is to solve for the N transmission linesusing the existing terminations with no coupling, i.e., N(ω)=0. The netresult of this difference between the initial-value problem in theVolterra iteration and the boundary-value problem of the telegrapher'sequation is that the traditional analysis of the Picard iteration andits proof of convergence to the unique solution of the Volterra integralequation does not apply to the transmission-line case. Nonetheless,knowledge of the “initial vector” Φ(0,ω) can be dispensed with ifinstead of writing the integral equation of currents and voltages as inEquation 8, the equation of the near-end/far-end transfer matrix of themulticonductor transmission lines is written.

Volterra Integral Equation for the Transfer Matrix

It is well known that this transfer matrix T(x, ω) is an exponentialmatrix that satisfies the ordinary matrix differential equation:$\begin{matrix}{{\frac{\partial}{\partial x}{T\left( {x,\omega} \right)}} = {{- \left( {{D(\omega)} + {N(\omega)}} \right)}{T\left( {x,\omega} \right)}}} & (11)\end{matrix}$with the initial condition T(x,ω)=I, where I is the 2N×2N identitymatrix. This equation can be written as: $\begin{matrix}{{\frac{\partial}{\partial x}{T\left( {x,\omega} \right)}} = {{{- {D(\omega)}}{T\left( {x,\omega} \right)}} - {{N(\omega)}{T\left( {x,\omega} \right)}}}} & (12)\end{matrix}$where the term N(ω)T(x, ω) is considered a forcing input. The generalsolution of the above equation is written as:T(x,ω)=e ^(−D(ω)x) −∫ ^(x) ₀ e ^(−D(ω)(x−ζ)) N(ω)T(ζ,ω)dζ  (13)

This matrix integral equation is indeed a Volterra integral equation ofthe second kind. It can be solved using the following iterative scheme:

Picard iteration for the transfer matrix of the multiconductortransmission line is the presence of off-diagonal splitting for the PULmatrices, as follows:

1. Set T⁽⁰⁾(x, ω)=e^(−D(ω)x.)

2. UpdateT(^(r+1))(x, ω)=e ^(−D(ω)x) +∫ ^(x) ₀ e ^(−D(ω)(x−ω))(−N(ω)T^((r))(ζ,w)dζ  (14).

3. Repeat (2) until convergence.

The properties of the Picard iteration are now used to derive anexplicit expression for the solution of Equation 12.

First denote by S^((r+1))(x, ω)=T^((r+1))(x, ω)−T^((r))(x, ω). Thenbased on the Picard iteration 14:S ^((r+1))(x,ω)=∫^(x) ₀ e ^(−D(ω)(x−ζ))(−N(ω))S ^((r))(ζ,ω)dζ  (15)which is the recursive equation defining all the S^((r))(x,ω) iterates.Taking the matrix norm of both sides, the following inequalities areobtained: $\begin{matrix}\begin{matrix}{{{S^{({r\quad + \quad 1})}\left( {x,\omega} \right)}} = {{\int_{0}^{x}{{{\mathbb{e}}^{- D_{{(\omega)}{({x - \zeta})}}}\left( {- {N(\omega)}} \right)}\quad{S^{(r)}\left( {\zeta,\omega} \right)}{\mathbb{d}\zeta}}}}} \\{\leq {\int_{0}^{x}{{{{{\mathbb{e}}^{- D_{{(\omega)}{({x - \zeta})}}}\left( {- {N(\omega)}} \right)}\quad{S^{(r)}\left( {\zeta,\omega} \right)}}}{\mathbb{d}\zeta}}}} \\{\leq {\int_{0}^{x}{{{\mathbb{e}}^{- D_{{(\omega)}{({x - \zeta})}}}}{\left( {- {N(\omega)}} \right)}{{S^{(r)}\left( {\zeta,\omega} \right)}}{\mathbb{d}\zeta}}}} \\{\leq {{{N(\omega)}}{\int_{0}^{x}{{{\mathbb{e}}^{- D_{{(\omega)}{({x - \zeta})}}}}{{S^{(r)}\left( {\zeta,\omega} \right)}}{\mathbb{d}\zeta}}}}}\end{matrix} & (16)\end{matrix}$

Since the function (x,ζ)−

→∥e^(−D(ω)(x−ζ)∥ is continuous on the square [0,L]×[0,L], where L is thelength of the transmission line, it has an upper bound denoted by E(ω).Note also that ∥e^(−D(ω)(x))∥≦E(ω). $\begin{matrix}{{{{S^{(r)}\left( {x,\omega} \right)}} \leq {\frac{x^{r}}{r^{!}}{{{N(\omega)}}^{r}\left\lbrack {E(\omega)} \right\rbrack}^{r + 1}}},{\forall{x \in \left\lbrack {0,L} \right\rbrack}}} & (17) \\{{{{S^{(r)}\left( {x,\omega} \right)}} \leq {\frac{L^{r}}{r!}{{{N(\omega)}}^{r}\left\lbrack {E(\omega)} \right\rbrack}^{r + 1}}},{\forall{x \in \left\lbrack {0,L} \right\rbrack}}} & (18)\end{matrix}$

The upper bound of 18 proves that the series${T^{(r)}\left( {x,\omega} \right)} = {\sum\limits_{k = 0}^{r}{S^{(k)}\left( {x,w} \right)}}$with S⁽⁰⁾(x, ω)=T⁽⁰⁾(x, ω) coverges uniformly on the interval [0,L] to alimit F(x, ω). It can be shown that this matrix function is indeed thesolution of the Volterra integral equation 13.

Error Analysis of the TWR Algorithm

The results of the previous section allows a rigorous error analysis ofthe transversal WR algorithm. As discussed hereinafter, the TWRgenerates the correct solution incrementally as a convergent series(every iteration one term is added to the series). The size of everyterm can be quantified. An upper bound on the error when the series istruncated will be derived.

When applied to multiconductor transmission lines, the residual errorR^((n)) (x, ω) after n iterations is given by $\begin{matrix}\begin{matrix}{{R^{(n)}\left( {x,\omega} \right)} = {{T^{(n)}\left( {x,\omega} \right)} - {\sum\limits_{k = 0}^{n}{S^{(k)}\left( {x,\omega} \right)}}}} \\{= {\sum\limits_{k = 0}^{\infty}{S^{(k)}\left( {x,\omega} \right)}}} \\{= {\sum\limits_{k = 0}^{r}{S^{(n)}\left( {x,\omega} \right)}}} \\{= {\sum\limits_{k = {n + 1}}^{\infty}{S^{(k)}\left( {x,\omega} \right)}}}\end{matrix} & (19)\end{matrix}$

Taking the matrix norms of both sides and using the upper bound of 18,the following (absolute) upper bound is obtained on the residual error:$\begin{matrix}\begin{matrix}{{{R^{(n)}\left( {x,\omega} \right)}} \leq {\sum\limits_{k = {n + 1}}^{\infty}{\frac{L^{k}}{k!}{{{N(\omega)}}^{k}\left\lbrack {E(\omega)} \right\rbrack}^{k + 1}}}} \\{= {\frac{L^{n + 1}}{\left( {n + 1} \right)!}{{{N(\omega)}}^{n + 1}\left\lbrack {E(\omega)} \right\rbrack}^{n + 2}{\sum\limits_{k = 0}^{\infty}{\frac{L^{k}}{k!}{{{N(\omega)}}^{k}\left\lbrack {E(\omega)} \right\rbrack}^{k}}}}} \\{= {\frac{L^{n + 1}}{\left( {n + 1} \right)!}{{{N(\omega)}}^{n + 1}\left\lbrack {E(\omega)} \right\rbrack}^{n + 2}\quad{\exp\left( {L{{N(\omega)}}{E(\omega)}} \right)}}}\end{matrix} & (20)\end{matrix}$where n is the iteration number, L is the line length, ∥N(ω)∥ measuresthe size of the coupling terms, and E(ω) indicates the (intrinsic)single line bound. These quantities from equation 20 can be computed apriori based on the input data.

The exponential in the final inequality is independent of the iterationn and is in fact an upper bound on the matrix norm ∥T(x,ω)∥. An estimateof the maximum relative error ρ_(n)(ω) of the WR approximation of themulticonductor transmission lines can be given by: $\begin{matrix}\begin{matrix}{{\rho_{n}(\omega)} = {{\exp\left( {{- L}{{N(\omega)}}{E(\omega)}} \right)}\quad{\max_{x \in {\lbrack{0,L}\rbrack}}{{R^{(n)}\left( {x,\omega} \right)}}}}} \\{\leq {\frac{L^{n + 1}}{\left( {n + 1} \right)!}{{{N(\omega)}}^{n + 1}\left\lbrack {E(\omega)} \right\rbrack}^{n + 2}}}\end{matrix} & (21)\end{matrix}$

The matrix norm ∥N(ω)∥ measures the size of the coupling terms in thePUL matrices. This relative error estimate allows a prediction that theweaker the couplings the faster the convergence. Furthermore, the longerthe line, the slower the convergence. These predictions are borne out byrecent computational evidence.

In order to evaluate E(ω)=max_(xε[0,L])∥exp(−D(ω)x)∥, the exponentialmatrix exp (∥D(ω)x) must be computed, where: $\begin{matrix}{{{D(\omega)}\begin{bmatrix}O & {D_{Z}(\omega)} \\{D_{Y}(\omega)} & O\end{bmatrix}} = \begin{bmatrix}O & {{diag}\left( {R_{\quad{ll}} + {j\quad\omega\quad L_{\quad{ll}}}} \right)} \\{{diag}\left( {G_{ll} + {j\quad\omega\quad C_{ll}}} \right)} & O\end{bmatrix}} & (22)\end{matrix}$

One way to compute the matrix exponential is by using the eigenanalysisof D(ω). For is there is an invertible matrix M such the D(ω)=M⁻¹Λ(ω)Mthen exp(−D(ω)x)=M⁻¹ exp(−Λ(ω)x)M. Note that the square of the matrixD(ω) is diagonal: $\begin{matrix}\begin{matrix}{{D^{2}(\omega)} = \begin{bmatrix}{{D_{Z}(\omega)}{D_{Y}(\omega)}} & O \\O & {{D_{Y}(\omega)}{D_{Z}(\omega)}}\end{bmatrix}} \\{= \begin{bmatrix}{\Gamma^{2}(\omega)} & O \\O & {\Gamma^{2}(\omega)}\end{bmatrix}}\end{matrix} & (23)\end{matrix}$where Γ²(ω)=diag(γ_(∥) ²(ω))=diag((R_(∥)+jωL_(∥))(G_(∥)+jωC_(∥))). Ifeach γ_(∥)(ω)=√{square root over((R_(∥)+jωL_(∥))(G_(∥)+jωC_(∥)))}=α_(∥)(ω)+jβ_(∥)(ω) is written, whereα_(∥)(ω) is the attenuation constant and β_(∥)(ω) is the phase constant,then the spectral radius of exp(−D(ω)x) will be max, exp(α_(∥)(ω)x).

FIG. 4 is a flow chart describing an exemplary Transversal WaveformRelaxation process 400 that incorporates a stopping criterion inaccordance with the present invention. As shown in FIG. 4, the exemplarytransversal waveform relaxation process 400 initially obtains the R, L,G, C parameters during step 410 for the multiconductor transmission linesystem under consideration. Thereafter, during step 420, the transversalwaveform relaxation process 400 separates the R, L, G, C matrices intotheir respective diagonal and non-diagonal portions, corresponding tointrinsic and coupled characteristics, respectively.

During step 430, the transversal waveform relaxation process 400computes the intrinsic behavior, E(ω), and strength of coupling, N(ω).The line length is obtained during step 440. The relative error isobtained during step 450 using equation (21). The error threshold isobtained during step 460, and compared during step 470 to the relativeerror calculated during step 450 until the threshold is reached. Oncethe error threshold is satisfied, the circuit simulation is generatedduring step 480 and the results are generated during step 490.

While exemplary embodiments of the present invention have been describedwith respect to digital logic blocks, as would be apparent to oneskilled in the art, various functions may be implemented in the digitaldomain as processing steps in a software program, in hardware by circuitelements or state machines, or in combination of both software andhardware. Such software may be employed in, for example, a digitalsignal processor, micro-controller, or general-purpose computer. Suchhardware and software may be embodied within circuits implemented withinan integrated circuit.

Thus, the functions of the present invention can be embodied in the formof methods and apparatuses for practicing those methods. One or moreaspects of the present invention can be embodied in the form of programcode, for example, whether stored in a storage medium, loaded intoand/or executed by a machine, or transmitted over some transmissionmedium, wherein, when the program code is loaded into and executed by amachine, such as a computer, the machine becomes an apparatus forpracticing the invention. When implemented on a general-purposeprocessor, the program code segments combine with the processor toprovide a device that operates analogously to specific logic circuits.

System and Article of Manufacture Details

As is known in the art, the methods and apparatus discussed herein maybe distributed as an article of manufacture that itself comprises acomputer readable medium having computer readable code means embodiedthereon. The computer readable program code means is operable, inconjunction with a computer system, to carry out all or some of thesteps to perform the methods or create the apparatuses discussed herein.The computer readable medium may be a recordable medium (e.g., floppydisks, hard drives, compact disks, , memory cards, semiconductordevices, chips, application specific integrated circuits (ASICs)) or maybe a transmission medium (e.g., a network comprising fiber-optics, theworld-wide web, cables, or a wireless channel using time-divisionmultiple access, code-division multiple access, or other radio-frequencychannel). Any medium known or developed that can store informationsuitable for use with a computer system may be used. Thecomputer-readable code means is any mechanism for allowing a computer toread instructions and data, such as magnetic variations on a magneticmedia or height variations on the surface of a compact disk.

The computer systems and servers described herein each contain a memorythat will configure associated processors to implement the methods,steps, and functions disclosed herein. The memories could be distributedor local and the processors could be distributed or singular. Thememories could be implemented as an electrical, magnetic or opticalmemory, or any combination of these or other types of storage devices.Moreover, the term “memory” should be construed broadly enough toencompass any information able to be read from or written to an addressin the addressable space accessed by an associated processor. With thisdefinition, information on a network is still within a memory becausethe associated processor can retrieve the information from the network.

It is to be understood that the embodiments and variations shown anddescribed herein are merely illustrative of the principles of thisinvention and that various modifications may be implemented by thoseskilled in the art without departing from the scope and spirit of theinvention.

1. A method for analyzing a circuit with transmission lines, comprising:obtaining one or more transmission line parameters of said circuit;obtaining intrinsic behavior, E(ω), and strength of coupling, N(ω), ofeach of said transmission lines; obtaining a relative error bound forsaid circuit based on said intrinsic behavior, E(ω), and strength ofcoupling, N(ω), of said transmission lines; obtaining a predefined errorthreshold; and iterating until said relative error bound satisfies saiderror threshold.
 2. The method of claim 1, wherein said one or moretransmission line parameters of said circuit comprises one or more ofcapacitance, resistance, inductance and conductance.
 3. The method ofclaim 1, wherein said relative error bound, ρ_(n)(ω), is expressed as:$\begin{matrix}{{\rho_{n}(\omega)} = {{\exp\left( {{- L}{{N(\omega)}}{E(\omega)}} \right)}\quad{\max_{x \in {\lbrack{0,L}\rbrack}}{{R^{(n)}\left( {x,\omega} \right)}}}}} \\{\leq {\frac{L^{n + 1}}{\left( {n + 1} \right)!}{{{N(\omega)}}^{n + 1}\left\lbrack {E(\omega)} \right\rbrack}^{n + 2}}}\end{matrix}$ where n is the iteration number and L is the line lengthof said transmission lines.
 4. The method of claim 1, wherein saidintrinsic behavior, E(ω), indicates the single line bound of eachtransmission line.
 5. The method of claim 1, wherein said strength ofcoupling, N(ω), measures the size of the coupling terms between two ormore transmission lines.
 6. The method of claim 1, further comprisingthe step of computing coupling factors for each transmission line basedon neighboring sources.
 7. The method of claim 1, further comprising thestep of modeling the transmission lines in terms of voltage and/orcurrent sources.
 8. An apparatus analyzing a circuit with transmissionlines, comprising: a memory; and at least one processor, coupled to thememory, operative to: obtain one or more transmission line parameters ofsaid circuit; obtain intrinsic behavior, E(ω), and strength of coupling,N(ω), of each of said transmission lines; obtain a relative error boundfor said circuit based on said intrinsic behavior, E(ω), and strength ofcoupling, N(ω), of said transmission lines; obtain a predefined errorthreshold; and iterate until said relative error bound satisfies saiderror threshold.
 9. The apparatus of claim 8, wherein said one or moretransmission line parameters of said circuit comprises one or more ofcapacitance, resistance, inductance and conductance.
 10. The apparatusof claim 8, wherein said relative error bound, ρ_(n)(ω), is expressedas: $\begin{matrix}{{\rho_{n}(\omega)} = {{\exp\left( {{- L}{{N(\omega)}}{E(\omega)}} \right)}\quad{\max_{x \in {\lbrack{0,L}\rbrack}}{{R^{(n)}\left( {x,\omega} \right)}}}}} \\{\leq {\frac{L^{n + 1}}{\left( {n + 1} \right)!}{{{N(\omega)}}^{n + 1}\left\lbrack {E(\omega)} \right\rbrack}^{n + 2}}}\end{matrix}$ where n is the iteration number and L is the line lengthof said transmission lines.
 11. The apparatus of claim 8, wherein saidintrinsic behavior, E(ω), indicates the single line bound of eachtransmission line.
 12. The apparatus of claim 8, wherein said strengthof coupling, N(ω), measures the size of the coupling terms between twoor more transmission lines.
 13. The apparatus of claim 8, wherein saidprocessor is further configured to compute coupling factors for eachtransmission line based on neighboring sources.
 14. The apparatus ofclaim 8, wherein said processor is further configured to model thetransmission lines in terms of voltage and/or current sources.
 15. Anarticle of manufacture for analyzing a circuit with transmission lines,comprising a machine readable medium containing one or more programswhich when executed implement the steps of: obtaining one or moretransmission line parameters of said circuit; obtaining intrinsicbehavior, E(ω), and strength of coupling, N(ω), of each of saidtransmission lines; obtaining a relative error bound for said circuitbased on said intrinsic behavior, E(ω), and strength of coupling, N(ω),of said transmission lines; obtaining a predefined error threshold; anditerating until said relative error bound satisfies said errorthreshold.
 16. The article of manufacture of claim 15, wherein said oneor more transmission line parameters of said circuit comprises one ormore of capacitance, resistance, inductance and conductance.
 17. Thearticle of manufacture of claim 15, wherein said relative error bound,ρ_(n)(ω), is expressed as: $\begin{matrix}{{\rho_{n}(\omega)} = {{\exp\left( {{- L}{{N(\omega)}}{E(\omega)}} \right)}\quad{\max_{x \in {\lbrack{0,L}\rbrack}}{{R^{(n)}\left( {x,\omega} \right)}}}}} \\{\leq {\frac{L^{n + 1}}{\left( {n + 1} \right)!}{{{N(\omega)}}^{n + 1}\left\lbrack {E(\omega)} \right\rbrack}^{n + 2}}}\end{matrix}$ where n is the iteration number and L is the line lengthof said transmission lines.
 18. The article of manufacture of claim 15,wherein said intrinsic behavior, E(ω), indicates the single line boundof each transmission line.
 19. The article of manufacture of claim 15,wherein said strength of coupling, N(ω), measures the size of thecoupling terms between two or more transmission lines.
 20. The articleof manufacture of claim 15, further comprising the step of computingcoupling factors for each transmission line based on neighboringsources.